# Tagged Questions

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### Can every discrete martingale be embedded in a continuous martingale?

Let $(X_k)_{k=0,1,..., n}$ be a discrete martingale defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$. I would like to know whether there exists a (continuous) martingale ...
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### The jump and the left martingale of semimartingale

Let $X_{t}$ be a semimartingale. Define $\Delta X_{t} = X_{t}- X_{t-}$. For fixed $s> 0$, $\Delta X_{s}$ and $X_{s-}$ are two random variable. Are they independent to each other? I think the ...
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### The regularity of Levy process

There is a property for continuous Markov process that each point $y$ in its state space is hit with positive probability one starting from any interior point $x$. This property is called the ...
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### The distribution of Jump gaps of Levy process

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...
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### Feynman-Kac for jump-diffusion

I'm looking for a more general Feynman-Kac formula that works in the case of jump-diffusion processes. I know that, given a pure diffusion process like $$dS_t=\mu_tdt+\sigma_tdW_t,$$ if $u(t,s)$ ...
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### Asymptotic behavior of solutions of stochastic differential equations

I am studying a risk model whose dynamic is specified by a first order differential equation with a compound Poisson process on the right hand side. I would like to know whether there are some papers ...
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### Supermartingale inequality on a particular event

Say, I have a supermartingale $Y_t$ with respect to the filtration $F_t$. Let $T$ and $S$ two stopping times greater than $t>0$ such that on the event $A$, $T>S$, then since $Y_t$ is a ...
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### Solving a SDE with quadratic drift

I am wondering whether the following SDE can be solved explicitly? $$d X_t = X_t^2 d t + X_t d B_t$$ where $B_t$ is a standard Brownian motion. If not, can we say some thing about the moments of ...
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### Existence of predictable quadratic covariation for a special pair of local martingales

In Limit theorems for stochastic processes, by Jacod and Shiryaev we have the existence of a predictable quadratic covariation process stated as the following theorem $\mathbf{Theorem}$ To each ...
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### Colored noise in SDE

I want to numerically study the behavior of a system described by a set of differential equations in the presence of colored noise. It seems that the standard procedure is to use the Langevin ...
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### Conditional law of an Ito's stochastic integral

Consider $B=(B_t)_{t\geq 0}$ real $\mathcal F_t$ - brownian motion starting at zero, in a probability space $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P)$. Then, consider a new real ...
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### Ito formulae for stochastic processes with finite cubic, quartic … n-tic variation

Many stochastic processes that you encounter are kind of well-behaved, i.e. have infinite variation, yet finite quadratic variation. My question revolves around stochastic processes that have ...
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### Upper bound concerning Snell envelope

Consider, on a filtred probability space $\left (\Omega, \mathcal F, \mathbb F , \mathbb P \right )$ where $\mathbb F = \left(\mathcal F_ t \right )_ {t\geq 0}$ is filtration satisfying the usuual ...
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### Parameter Sensitivity of Stochastic Process

How do I compute the derivative \frac{\partial X_t}{\partial \sigma}? Where dX_t=\theta (\mu-X_t)dt+\sigma \sqrt{X_t}dZ_t